Random matrices, log-gases and Hölder regularity Conference Paper


Author(s): Erdős, László
Title: Random matrices, log-gases and Hölder regularity
Title Series: Proceedings of the International Congress of Mathematicians
Affiliation IST Austria
Abstract: The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner’s original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices.
Keywords: De Giorgi-Nash-Moser parabolic regularity, Wigner-Dyson-Gaudin-Mehta universality, Dyson Brownian motion
Conference Title: ICM: International Congress of Mathematicians
Volume: 3
Conference Dates: August 13 - 21, 2014
Conference Location: Seoul, Korea
ISBN: 978-89-6105-806-3
Publisher: Kyung Moon SA Co. Ltd.  
Location: Seoul
Date Published: 2014-08-01
Start Page: 214
End Page: 236
Sponsor: The author is partially supported by SFB-TR 12 Grant of the German Research Council and by ERC Advanced Grant, RANMAT 338804.
URL:
Open access: yes (repository)
IST Austria Authors
  1. László Erdős
    110 Erdős
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